I guess the difference between these terms comes from what they do, some computer specific differences. "Rechtschreibkontrolle " is desgined for interaction between user and computer. the user works on "Rechtschreibprüfung" and "Rechtschreibkontrolle " checks the spelling with some internal dictionaries and user see the results of this process in "Rechtschreibprüfung".

die Rechtschreibprüfung is the program's user interface which allow users to approve replacements and modify the program's operation.